oppc1stf - Mathbox for Zhi Wang

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Theorem oppc1stf 49946

Description: The opposite functor of the first projection functor is the first projection functor of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.)

**Hypotheses**
Ref	Expression
oppc1stf.o	⊢ 𝑂 = (oppCat‘𝐶)
oppc1stf.p	⊢ 𝑃 = (oppCat‘𝐷)
oppc1stf.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
oppc1stf.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

**Assertion**
Ref	Expression
oppc1stf	⊢ (𝜑 → ( oppFunc ‘(𝐶 1^st_F 𝐷)) = (𝑂 1^st_F 𝑃))

Proof of Theorem oppc1stf Dummy variables 𝑥 𝑦 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppc1stf.o	. 2 ⊢ 𝑂 = (oppCat‘𝐶)
2		oppc1stf.p	. 2 ⊢ 𝑃 = (oppCat‘𝐷)
3		oppc1stf.c	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
4		oppc1stf.d	. 2 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
5		eqid 2769	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))
6	5	tposmpo 8255	. . . . 5 ⊢ tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))) = (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))
7		eqid 2769	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8	7, 1	oppchom 17767	. . . . . . . . 9 ⊢ ((1^st ‘𝑦)(Hom ‘𝑂)(1^st ‘𝑥)) = ((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦))
9		eqid 2769	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	9, 2	oppchom 17767	. . . . . . . . 9 ⊢ ((2^nd ‘𝑦)(Hom ‘𝑃)(2^nd ‘𝑥)) = ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦))
11	8, 10	xpeq12i 5687	. . . . . . . 8 ⊢ (((1^st ‘𝑦)(Hom ‘𝑂)(1^st ‘𝑥)) × ((2^nd ‘𝑦)(Hom ‘𝑃)(2^nd ‘𝑥))) = (((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦)))
12		eqid 2769	. . . . . . . . 9 ⊢ (𝑂 ×_c 𝑃) = (𝑂 ×_c 𝑃)
13		eqid 2769	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
14	1, 13	oppcbas 17770	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝑂)
15		eqid 2769	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
16	2, 15	oppcbas 17770	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝑃)
17	12, 14, 16	xpcbas 18230	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝑂 ×_c 𝑃))
18		eqid 2769	. . . . . . . . 9 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
19		eqid 2769	. . . . . . . . 9 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
20		eqid 2769	. . . . . . . . 9 ⊢ (Hom ‘(𝑂 ×_c 𝑃)) = (Hom ‘(𝑂 ×_c 𝑃))
21		simp2 1153	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))
22		simp3 1154	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
23	12, 17, 18, 19, 20, 21, 22	xpchom 18232	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥) = (((1^st ‘𝑦)(Hom ‘𝑂)(1^st ‘𝑥)) × ((2^nd ‘𝑦)(Hom ‘𝑃)(2^nd ‘𝑥))))
24		eqid 2769	. . . . . . . . 9 ⊢ (𝐶 ×_c 𝐷) = (𝐶 ×_c 𝐷)
25	24, 13, 15	xpcbas 18230	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×_c 𝐷))
26		eqid 2769	. . . . . . . . 9 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
27	24, 25, 7, 9, 26, 22, 21	xpchom 18232	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦) = (((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦))))
28	11, 23, 27	3eqtr4a 2830	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥) = (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))
29	28	reseq2d 5976	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥)) = (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))
30	29	mpoeq3dva 7485	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥))) = (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))))
31	6, 30	eqtr4id 2823	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))) = (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥))))
32	31	opeq2d 4846	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))⟩ = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥)))⟩)
33		simprl 782	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝐶 ∈ Cat)
34		simprr 784	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝐷 ∈ Cat)
35		eqid 2769	. . . . 5 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
36	24, 25, 26, 33, 34, 35	1stfval 18243	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝐶 1^st_F 𝐷) = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))⟩)
37	24, 33, 34, 35	1stfcl 18249	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝐶 1^st_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐶))
38	36, 37	oppfval3 49796	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ( oppFunc ‘(𝐶 1^st_F 𝐷)) = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))⟩)
39	1	oppccat 17774	. . . . 5 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
40	33, 39	syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝑂 ∈ Cat)
41	2	oppccat 17774	. . . . 5 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
42	34, 41	syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝑃 ∈ Cat)
43		eqid 2769	. . . 4 ⊢ (𝑂 1^st_F 𝑃) = (𝑂 1^st_F 𝑃)
44	12, 17, 20, 40, 42, 43	1stfval 18243	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝑂 1^st_F 𝑃) = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥)))⟩)
45	32, 38, 44	3eqtr4d 2814	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ( oppFunc ‘(𝐶 1^st_F 𝐷)) = (𝑂 1^st_F 𝑃))
46		df-1stf 18225	. 2 ⊢ 1^st_F = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⦋((Base‘𝑐) × (Base‘𝑑)) / 𝑏⦌⟨(1^st ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1^st ↾ (𝑥(Hom ‘(𝑐 ×_c 𝑑))𝑦)))⟩)
47	1, 2, 3, 4, 45, 46	oppc1stflem 49945	1 ⊢ (𝜑 → ( oppFunc ‘(𝐶 1^st_F 𝐷)) = (𝑂 1^st_F 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ⦋csb 3861 ⟨cop 4597 × cxp 5657 ↾ cres 5661 ‘cfv 6534 (class class class)co 7408 ∈ cmpo 7410 1^st c1st 7980 2^nd c2nd 7981 tpos ctpos 8217 Basecbs 17265 Hom chom 17317 Catccat 17716 oppCatcoppc 17763 ×_c cxpc 18220 1^st_F c1stf 18221 oppFunc coppf 49780

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-z 12588 df-dec 12708 df-uz 12859 df-fz 13532 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-hom 17330 df-cco 17331 df-cat 17720 df-cid 17721 df-homf 17722 df-comf 17723 df-oppc 17764 df-func 17911 df-xpc 18224 df-1stf 18225 df-oppf 49781

This theorem is referenced by: (None)

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